1. CHAPTER 8. Approximation Theory
Dongshin Kim
Computer Networks Research Lab.
Dept. of Computer Science and Engineering
Korea University
9 november 2005.

2. Contents Discrete Least Squares Approximation
Orthogonal Polynomials and Least Squares Approximation
Chebyshev Polynomials and Economization of Power Series
Rational Function Approximation
Trigonometric Polynomial Approximation
Fast Fourier Transforms

3. Discrete Least Squares Approximation
Finding best equation
minimize a0 and a1
Another approach (absolute deviation):

4. Discrete Least Squares Approximation

5. Discrete Least Squares Approximation
Solution

6. Orthogonal Polynomials and Least Squares Approximation
Polynomial Pn(x)

7. Orthogonal Polynomials and Least Squares Approximation
Definition 8.1: linearly independent
If is a polynomial of degree j, for each j=0,1,…,n, then is linearly independent on any interval [a, b]

8. Orthogonal Polynomials and Least Squares Approximation

9. Orthogonal Polynomials and Least Squares Approximation
Gram-Schmidt process

11. Orthogonal Polynomials and Least Squares Approximation

12. Chebyshev Polynomials and Economization of Power Series
Orthogonal on (-1,1) with respect to the weight function

13. Chebyshev Polynomials and Economization of Power Series

14. Chebyshev Polynomials and Economization of Power Series
Approximating an arbitrary nth-degree polynomial

15. Rational Function Approximation
Pade Rational Approximation

16. Rational Function Approximation

17. Result
n=5, m=0 p= q=
n=4, m=1 p= q=
n=3, m=2 p= q=

18. Result r(x) = p(x)/q(x) n=5, m=0 n=4, m=1 n=3, m=2
p(x)= *x *x^ *x^ *x^ *x^5
q(x)=
n=4, m=1
p(x)= *x *x^ *x^ *x^4
q(x)= *x
n=3, m=2
p(x)= *x *x^ *x^3
q(x)= *x *x^2

19. Graph

20. Result [f(x)-r(x)] n=5, m=0 n=4, m=1 n=3, m=2
n=4, m=1
1.0e-003 *
n=3, m=2
1.0e-004 *

21. Rational Function Approximation
Chebyshev Rational Approximation

22. Rational Function Approximation

23. Result
n=5, m=0 p= q=
n=4, m=1 p= q=
n=3, m=2 p= q=

24. Graph

25. Result [f(x)-r(x)] n=5, m=0 n=4, m=1 n=3, m=2 1.0e-004 *
n=4, m=1
1.0e-005 *
n=3, m=2

26. Result [f(x)-r(x)] Pade Chebyshev n=5, m=0 n=4, m=1 n=3, m=2
n=4, m=1
1.0e-003 *
n=3, m=2
1.0e-004 *
Chebyshev
1.0e-005 *

27. Trigonometric Polynomial Approximation
All linear combinations of the functions
Fourier series